Philosophical foundations of set theories of Georg Cantor and Petr Vopěnka

Considering infinite as actually infinite was denied in philosophy and science from the antiquity till 1870s. It sprang up as a result of Aristotelian analysis of Classical Greek notion omsipov, the sense of which was immensity or limitlessness, and was formalized in the formula Infinitum actu non datur. Georg Cantor introduced the concept of set in the middle of the 19 century, which represented assemblies of things as an individual object, particularly infinite assemblies. Cantor's set theory assumes that all its objects are formed at all; i.e., actually given. A Slovak mathematician, Petr Vopenka, proposed a crucially new sense of infinity - natural infinity - that is the result of considering of a sufficiently large set by a finite observer. This considering is characterised by increasing of unsharpness closer to horizon - the border, which constraints the view intended afield or inward if it has no clear obstruction. Unsharpness is not a disadvantage and allows to abstract wholeness from specialties, or to consider specialties, rejecting their integrality. Sharpness is a subcase of unsharpness. Horizon is not fixed and can move while observer is approaching. G. Cantor and P. Vopenka considered infinity from different points of view. So we need to answer a question: what is the cause of this difference? We can discover parts of ontological basis of G. Cantor's ideas in his works and authentically reconstruct it. He affirmed that mathematical objects have two types of reality - intrasubject (immanent) and transsubject (transient). Their connection is apodictical, and the immanent reality is primary so mathematics has to take into account only this type of reality. The foregoing allows us to determine the philosophical position of G. Cantor as a position of classical Platonism. In his works, P. Vopenka mentions about phenomenological rebuilding of the set theory and mathematics in whole. The main method is in the displacement of the investigators' point of view to the point of view of a finite observer, as contrary to God's point of view in Cantor's set theory. There is very frequent usage of words phenomenon and horizon in contexts and senses specific to phenomenology in Vopenka's texts. So we can conclude that philosophical position of P. Vopenka is very close to the Husserlian phenomenology. We come to the conclusion that the philosophical basis of Cantor's set theory is Platonism, whereas Vopenka's alternative set theory is founded on the Husserlian phenomenology; and this determines differences in the sense of infinity of the noted authors

Keywords

Authors

References